This paper proposes novel robust and low-overhead physical unclonable function (PUF) authentication and key exchange protocols that are resilient against reverse-engineering attacks. The protocols are executed between a party with access to a physical PUF (prover) and a trusted party who has access to the PUF compact model (verifier). The proposed protocols do not follow the classic paradigm of exposing the full PUF responses or a transformation of them. Instead, random subsets of the PUF response strings are sent to the verifier so the exact position of the subset is obfuscated for the third-party channel observers. Authentication of the responses at the verifier side is done by matching the substring to the available full response string; the index of the matching point is the actual obfuscated secret (or key) and not the response substring itself. We perform a thorough analysis of resiliency of the protocols against various adversarial acts, including machine learning and statistical attacks. The attack analysis guides us in tuning the parameters of the protocol for an efficient and secure implementation. The low overhead and practicality of the protocols are evaluated and confirmed by hardware implementation.